Graphing Rational Functions by Hand

Overview

• Objectives
• Reasons Why This is a Worthwhile Activity
• What Types of Rational Functions are We Graphing Here?
• Comments
• Next page for directions on how to actually graph rational functions

Objectives

*A, B, and C objectives below are objectives that most students have "had" and understand a some level (however, often understanding is weak). By going through this activity of learning how to graph rational functions by hand, the understanding of A, B, and C objectives are strengthened.

†The D objectives are objectives that the student will learn as a result of doing this activity (you probably do not know these objectives yet). These are useful objectives about functions, rational functions, and graphing.

1. *Basic and Intermediate Number Sense Concepts
2. *Basic Function Sense
3. *Procedural Skills
4. †Advanced Function Sense

1. *Basic and Intermediate Number Sense Concepts ~ The student should understand that:
   1. Zero times a number is zero.
   2. Zero plus a number is the number.
   3. A fraction is zero when the numerator (top) is zero.
   4. A fraction is undefined (not defined) when the denominator (bottom) is zero.
   5. pos./pos. = pos.; neg./neg. = pos.; neg./pos. = neg.; pos./neg. = neg. (where "pos." stands for a positive number, and "neg." stands for a negative number).
   6. When you divide a number by a small number (less than 1) the answer is bigger than the original number.
   7. When you divide a number by a tiny number (close to zero) the answer is large (the smaller the divisor, the larger the answer).

 

2. *Basic Function Sense ~ The student should understand that:

1. When plugging zero into a polynomial, you get the constant term.
2. To find the x-intercept, put zero in for y and solve.
3. To find the y-intercept, put zero in for x and solve.
4. If a polynomial is factored, and (x-a) is a factor, "a" being a number, then if you plug in x=a, then the result is zero.

 

3. *Procedural Skills ~ The student should be able to:

1. Plug zero into a polynomial that is in expanded form (use objective B.1).
2. Plug zero into a polynomial that is in factored form.
3. Plug zero into a rational function.

 

4. †Advanced Function Sense ~ The student should understand that:

1. When the denominator is zero (for a rational function) it usually means there is a vertical asymptote at the value that causes the denominator to be zero.
2. If (x-a) is a factor of the denominator, "a" being a number, if one puts in x-values very close to a, then the denominator is very small, usually causing the function to have a value which is large positive or large negative.

3. On either side of a vertical asymptote the graph commonly goes up "to positive infinity" or "down to negative infinity" (which means that the function values grow large without bound, or become largely negative without bound).

4. For a rational function, as the input values grow large ("approach positive or negative infinity"), the function values (value of the fraction) might do one of three things:

   1. Approach 0 (this happens when the denominator [bottom] dominates the fraction).
   2. Keep growing (positively or negatively - this happens when the numerator [top] dominates the fraction).
   3. Approach a specific number (this happens when neither the numerator or the denominator dominates the fraction).
5. For a polynomial, the term with the highest degree will dominate the polynomial.
6. For a rational function, the term with the highest degree will dominate the rational function, if there is one.

Reasons Why This is a Worthwhile Activity

1. The activity brings together number sense, function sense, and graph sense. The primary strength of the activity is the connections is builds.
2. While the process my appear procedural, it is largely a conceptual process of finding relationship--rather like putting a puzzle together.
3. Helps one realize that all functions are not nice lines or curves without breaks. Function graphs can have breaks.
4. Helps one consider "global" features of graphs, not just pointwise information.
5. By using basic skills as part of a bigger problem, one better learn the basic skills.
6. Students often have difficulty with asymptotes (not just the spelling). By graphing rational functions by hand, one learns what the asymptotes mean.
7. Helps one realize that sometimes one can learn more by graphing by hand than by using technology.

What Types of Rational Functions are We Graphing Here?

To keep the amount of arithmetic low, and allow for the graphs to be constructed conceptually, the following is true of the rational functions in this activity.

• Numerators and denominators are polynomials, frequently linear and with integer coefficients.
• Usually numerators and denominators of degree two or higher are factorable (or already factored).
• Generally, the pathological same-factor-in-both-the-numerator-and-denominator is not here (but that can be added later).

Comments

1. Generally, I am in favor of using a graphics calculator (or other technology) for graphing functions. However, in this case, producing by-hand graphs is highly desirable. They can be produced quite easily (with very little computational arithmetic) and the process of producing the graphs helps one understand many concepts related to functions.
2. Graphics calculators (graphs and tables) may be useful in the learning process. However, the intention is that they are only tools to help the student learn the concepts. Once the concepts are understood, the graphics calculator is no longer needed and should no longer be used.
3. Generally, I am in favor of lessons that have real-world applications. This lesson is a counterexample to that principle as well. This activity requires abstract thinking, without a lot of computational arithmetic.
4. Ironically, an activity that appears to be procedural in nature turns out to be quite conceptual.

Next page for directions on how to actually graph rational functions

Updated: March 19, 2008